Exercise 1, § 3

Let $A$ be a commutative Banach algebra and let $I$ be a maximal ideal of $A$. Now prove that there are only two possible cases: either $I$ is the kernel of a character of $A$, or $I$ is a hyperplane containing $A^2$. (Use Exercise 6 of I, p. 154). In particular, if $I$ is not closed, $I$ is a dense hyperplane of $A$ containing $A^2$. (To obtain an example of this latter situation, consider Exercise 3.)